The centre-back has fallen out of favour with Old Trafford boss Erik ten Hag, making only four league starts this season.
The Mirror reports United are prepared to make a huge loss on the defender they paid Leicester £85million for in 2019.
And it is claimed West Ham are ready to make a loan move for the unsettled England international.
Moyes once looked at signing Maguire for just £4m from Sheffield United while he was Old Trafford manager in 2013 but a possible move never took off.
United have won their last five league matches to rise to fourth in the table.
But Maguire has played little part, causing question marks over his future in Manchester.